On parabolic and umbilic points of immersed hypersurfaces

E. A. Feldman
1967 Transactions of the American Mathematical Society  
Introduction. This paper is a natural sequel to Geometry of immersions. I and II ([1] and [2]). The object is to apply the general geometric singularity theory developed in [1] and [2] to two specific geometric singularities of an immersed n manifold in Fn+1. These applications differ from those at the end of [2] as they involve a more detailed study of the singularity in question which sharpens the information given by the general theory. In §1 we review much of the necessary information from
more » ... y information from [1] and [2]. In §2 we set up the machinery for studying the parabolic points of an immersion of a compact n-manifold into Fn + 1 as a type of singularity of the osculating map (a refinement of the 2-jet). We also apply the main theory in §2. §3 refines the results of §2, especially in the case n = 2. In §4 and §5 we repeat the same process for umbilic points. We will close by illustrating how we proceed in the case of the umbilic points. Let Y be a compact 2-dimensional manifold. Let I{X, R3) be the set of immersions of X in R3. Let fe I{X, R3). We ask when if ever is the set of umbilics (points where the two principal curvatures are equal) a submanifold of codimension two, and if it is a submanifold in a particularly "nice" way, we ask how many umbilics there are. An application of the two main theorems [1] and [2] (and §1) gives the following answer. For an open dense subset Du of 7(X, R3), the number of umbilic points is even. This is roughly the content of §4. In §5 the transversality condition of "niceness" is studied in detail and is related to the index of an isolated umbilic point. The main result is: if/e Dn then the index of an umbilic point off is ±\, and therefore the number of umbilic points of/is greater than or equal to 2|x(Y)| where x(Y) is the Euler characteristic. A more complete outline of these ideas is found in the last sections of [3]. The material in §2 and §4 was contained in the author's doctoral dissertation at Columbia University. §3 and §5 are new. Finally, in addition to the people thanked in [1] and [2], I would like to thank J. Milnor for suggesting that 5.6 might be true. 0. Notations and conventions, (a) All manifolds discussed in this paper will be finite dimensional and all the connected components of a given manifold will have the same dimension. The manifolds will satisfy the second axiom of countability
doi:10.1090/s0002-9947-1967-0206974-1 fatcat:vo5vcfvc2jcq7pkuqq74fmu27q